“what do I wish I had been told (or was most useful from what I was told) when learning this material, and what should someone learning it today wish for in light of how things have developed”,

subject to any applicable restrictions on the amount/difficulty/scope of material that can be presented in the particular context. This is both deep and circular. Deep as a Zen or Occam’s Razor-like mantra to pervasively keep things honest. Circular in that it begs the question of what to value, which leads back to the starting point of making lists of explicit criteria to choose which things are more desirable than others.

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]]>The book [i]The Art of Combinatorial Proof[/i] comes to my mind. Proving a simple Fibonacci identity with tilings seems to be an enlightening exercise which boosts your appreciation both for the tiling interpretation of Fibonacci numbers and combinatorial proof in general. A specific example we proved (actually with Snake Oil, not tiling) in a class I helped with today is

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]]>What does “qualifications” entail? Contest math results/teaching experience/publications? High school transcripts?

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]]>With this kind of precision guided questioning, one may think as if someone is ready to launch a full fledged FBI investigation into author’s personal experiences; putting to alternate use the depth of experiences the gained from angel investing.

Nowhere did the author even remotely hint to pretend to suggest “failure of the educational system” or apparently display any trace of rancor of someone’s having been “failed by the system”. Then why this line of questioning?

There may indeed exist pockets of similar localized challenges that folks may have come across. I would simply read this blog as an honest, emotive expression of a somewhat imperfect experience being recounted by a talented mathematician dedicated to giving back to the subject outlining the rigor and perhaps extra support needed for potential excellence in Math or any subject for that matter.

On the other hand, I could relate to some of author’s experiences based on similar fun and joy what my close friend’s son is going through in his middle school right now. My friend will probably take to home schooling starting high school next year to help the child get immersed in math in the proverbial Einsteinian flow. I can understand it would be another manifestation of a personal choice, available to all, drastic to many. However, I am sure even they would not feel that the the “system failed them” if they made this decision. Similarly, on the converse side too.

Lastly, my thanks to the author for the article.

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]]>Your blog entry seems to be targeted specifically at exercises accompanying a lecture, i.e. exercises that students do while studying a new subject. Do you also have some insightful opinion on students that are already more experienced (like revisiting some subject)? So e.g. do you agree with the widespread opinion that one should do every exercise in Atiyah-Macdonald and Hartshorne and… somewhen if one wishes to go in that direction?

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]]>For those of us with worse rulers or who just suck at constructing perpendiculars, there’s an easier way to construct the orthocenter. Draw the A altitude and let it intersect BC at D and (ABC) at P. Then draw the circle with center D through P, and its other intersection with AD will be H (the orthocenter). This works because the reflection of H about BC lies on the circumcircle.

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]]>This is easier to prove when you prove it converges *absolutely* almost everywhere, and indeed one sees that if the original theorem is true, then so must be this absolute convergence theorem, since if each $f_i$ is replaced by its absolute value we can apply the theorem on the new sequence (the hypothesis is obviously still true) to get new sum function $g(x) = \sum_{i=1}^{\infty} |f_i|(x)$ which is guaranteed to converge almost everywhere; hence $f$ converges absolutely almost everywhere.

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